Assume that strong duality holds for the inner minimization problem and its Lagrangian dual (see, e.g., [1]}):
\(\forall a\in \mathcal {A}: \quad \min _{p \in P, f(a,p) \le 0} c^\top p = \max _{\lambda \succeq 0, \nu }g(a,\lambda ,\nu )\) 
